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I have a matrix Z n times p with p>n

I have A a diagonal matrix with positive entries

I would like to know if there is a knwon relation (as a function of A) between

the MP inverse of Z^T Z

and the MP inverse of A Z^T Z A

what i am looking for is the follwing: suppose I knwo the MP of $Z^T Z$ and I know $A$, can I get as a function of those two things the MP inverse of $A Z^T Z A$?

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# Moore-Penrose pseudo inverse

I have a matrix Z n times p with p>n

I have A a diagonal matrix with positive entries

I would like to know if there is a knwon relation (as a function of A) between

the MP inverse of Z^T Z

and the MP inverse of A Z^T Z A